Suggested flow: 6 minutes exploring, 5 building the matrix, 8 calculating eigenpairs, 5 converting frequencies, and 6 choosing an operating speed. Students work on paper before entering their answers. Later stages unlock after the preceding calculation is checked.
Discovery before explanation
Students start with both units at equilibrium and zero velocity. Have them switch on the motor, explore the whole 60–360 RPM range, and then refine two pronounced peaks. Record the larger maximum absolute displacement of either unit. Connection extension alone hides one of the modes, so it is not the exploration metric.
Every power or speed change resets the trial to equilibrium with the motor force at its positive maximum. This is a series of independent constant-speed tests, not a simulation of a motor accelerating between speeds. All reported maxima are over exactly 0–8 s; velocity zeros are numerically refined between graph samples.
The model and the minus sign
Each unit has mass 100 kg. Outer springs have stiffness 10,000 N/m and the connecting spring 40,000 N/m. Small horizontal displacements, ideal linear springs, and zero damping are assumed. Vertical weight is supported separately.
We define A as the stiffness matrix divided by mass. Then x″ + Ax = f(t)/100. Equivalently, x″ = −Ax + f(t)/100: the minus sign represents restoring spring forces. This choice makes each eigenvalue of A a squared angular natural frequency.
The motor applies 10 cos(Ωt) N to the left unit, with one force cycle per revolution. The force amplitude is held constant as RPM changes to isolate frequency effects. A real unbalanced rotor has a force amplitude that also depends on speed. The response is evaluated using the exact modal solution. At resonance the undamped model has a growing envelope; real damping limits growth, and real travel limits can invalidate the linear model.
Checks and interpretation
Approximate experimental speeds are accepted within 5 RPM of the two main eight-second peaks. The maxima of a finite test can sit slightly away from the theoretical natural frequencies. Eigenpairs can be entered in either order, with any nonzero scaling of each eigenvector. Decimal and fractional components are accepted. No eigenpair answers are displayed before students submit them.
The final task uses an explicitly chosen classroom requirement: 120–320 RPM operation, at least 15 RPM from each predicted resonance, and no more than 5 mm displacement of either unit over the eight-second test. These conditions support a model-based recommendation for that test. They do not establish performance for every duration, startup transient, harmonic, or physical machine.
Motor and controller choice
A programmable drive can select operating speeds and avoid lingering near resonances. A stepper motor is one possible implementation: pulse rate controls speed, but the stepper and its load can have resonances of their own. Real design also checks stepping harmonics, torque, acceleration through sensitive speeds, and measured damping.
Discussion prompts
- Why does a force on the left unit also move the right unit?
- What changes when we describe motion in an eigenvector basis?
- Why might an eight-second peak differ from a natural-frequency prediction?
- What additional measurements would support a final motor selection?
Background reading
MathWorks: Modal analysis · Oriental Motor: Stepper motor basics · Oriental Motor: Minimizing stepper vibration
Responses and trials are saved only in this browser. Print the notebook for submission. The checks support learning; they are not a secure assessment system.